1: *> \brief \b ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER IHI, ILO, INFO, LDA, N
25: * ..
26: * .. Array Arguments ..
27: * COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
37: *> by a unitary similarity transformation: Q**H * A * Q = H .
38: *> \endverbatim
39: *
40: * Arguments:
41: * ==========
42: *
43: *> \param[in] N
44: *> \verbatim
45: *> N is INTEGER
46: *> The order of the matrix A. N >= 0.
47: *> \endverbatim
48: *>
49: *> \param[in] ILO
50: *> \verbatim
51: *> ILO is INTEGER
52: *> \endverbatim
53: *>
54: *> \param[in] IHI
55: *> \verbatim
56: *> IHI is INTEGER
57: *>
58: *> It is assumed that A is already upper triangular in rows
59: *> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
60: *> set by a previous call to ZGEBAL; otherwise they should be
61: *> set to 1 and N respectively. See Further Details.
62: *> 1 <= ILO <= IHI <= max(1,N).
63: *> \endverbatim
64: *>
65: *> \param[in,out] A
66: *> \verbatim
67: *> A is COMPLEX*16 array, dimension (LDA,N)
68: *> On entry, the n by n general matrix to be reduced.
69: *> On exit, the upper triangle and the first subdiagonal of A
70: *> are overwritten with the upper Hessenberg matrix H, and the
71: *> elements below the first subdiagonal, with the array TAU,
72: *> represent the unitary matrix Q as a product of elementary
73: *> reflectors. See Further Details.
74: *> \endverbatim
75: *>
76: *> \param[in] LDA
77: *> \verbatim
78: *> LDA is INTEGER
79: *> The leading dimension of the array A. LDA >= max(1,N).
80: *> \endverbatim
81: *>
82: *> \param[out] TAU
83: *> \verbatim
84: *> TAU is COMPLEX*16 array, dimension (N-1)
85: *> The scalar factors of the elementary reflectors (see Further
86: *> Details).
87: *> \endverbatim
88: *>
89: *> \param[out] WORK
90: *> \verbatim
91: *> WORK is COMPLEX*16 array, dimension (N)
92: *> \endverbatim
93: *>
94: *> \param[out] INFO
95: *> \verbatim
96: *> INFO is INTEGER
97: *> = 0: successful exit
98: *> < 0: if INFO = -i, the i-th argument had an illegal value.
99: *> \endverbatim
100: *
101: * Authors:
102: * ========
103: *
104: *> \author Univ. of Tennessee
105: *> \author Univ. of California Berkeley
106: *> \author Univ. of Colorado Denver
107: *> \author NAG Ltd.
108: *
109: *> \ingroup complex16GEcomputational
110: *
111: *> \par Further Details:
112: * =====================
113: *>
114: *> \verbatim
115: *>
116: *> The matrix Q is represented as a product of (ihi-ilo) elementary
117: *> reflectors
118: *>
119: *> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
120: *>
121: *> Each H(i) has the form
122: *>
123: *> H(i) = I - tau * v * v**H
124: *>
125: *> where tau is a complex scalar, and v is a complex vector with
126: *> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
127: *> exit in A(i+2:ihi,i), and tau in TAU(i).
128: *>
129: *> The contents of A are illustrated by the following example, with
130: *> n = 7, ilo = 2 and ihi = 6:
131: *>
132: *> on entry, on exit,
133: *>
134: *> ( a a a a a a a ) ( a a h h h h a )
135: *> ( a a a a a a ) ( a h h h h a )
136: *> ( a a a a a a ) ( h h h h h h )
137: *> ( a a a a a a ) ( v2 h h h h h )
138: *> ( a a a a a a ) ( v2 v3 h h h h )
139: *> ( a a a a a a ) ( v2 v3 v4 h h h )
140: *> ( a ) ( a )
141: *>
142: *> where a denotes an element of the original matrix A, h denotes a
143: *> modified element of the upper Hessenberg matrix H, and vi denotes an
144: *> element of the vector defining H(i).
145: *> \endverbatim
146: *>
147: * =====================================================================
148: SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
149: *
150: * -- LAPACK computational routine --
151: * -- LAPACK is a software package provided by Univ. of Tennessee, --
152: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
153: *
154: * .. Scalar Arguments ..
155: INTEGER IHI, ILO, INFO, LDA, N
156: * ..
157: * .. Array Arguments ..
158: COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
159: * ..
160: *
161: * =====================================================================
162: *
163: * .. Parameters ..
164: COMPLEX*16 ONE
165: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
166: * ..
167: * .. Local Scalars ..
168: INTEGER I
169: COMPLEX*16 ALPHA
170: * ..
171: * .. External Subroutines ..
172: EXTERNAL XERBLA, ZLARF, ZLARFG
173: * ..
174: * .. Intrinsic Functions ..
175: INTRINSIC DCONJG, MAX, MIN
176: * ..
177: * .. Executable Statements ..
178: *
179: * Test the input parameters
180: *
181: INFO = 0
182: IF( N.LT.0 ) THEN
183: INFO = -1
184: ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
185: INFO = -2
186: ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
187: INFO = -3
188: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
189: INFO = -5
190: END IF
191: IF( INFO.NE.0 ) THEN
192: CALL XERBLA( 'ZGEHD2', -INFO )
193: RETURN
194: END IF
195: *
196: DO 10 I = ILO, IHI - 1
197: *
198: * Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
199: *
200: ALPHA = A( I+1, I )
201: CALL ZLARFG( IHI-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAU( I ) )
202: A( I+1, I ) = ONE
203: *
204: * Apply H(i) to A(1:ihi,i+1:ihi) from the right
205: *
206: CALL ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
207: $ A( 1, I+1 ), LDA, WORK )
208: *
209: * Apply H(i)**H to A(i+1:ihi,i+1:n) from the left
210: *
211: CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,
212: $ DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )
213: *
214: A( I+1, I ) = ALPHA
215: 10 CONTINUE
216: *
217: RETURN
218: *
219: * End of ZGEHD2
220: *
221: END
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